Quantum Physics

Sampling the Hydrogen Atom

Since the emergence of quantum theory just over a century ago every model for the hydrogen atom that has been developed incorporates the same basic assumption. From Niels Bohr through de Broglie and Schrödinger up to and including the Standard Model all such theories are based on an assumption put forward by John Nicholson. Nicholson was the first to recognise that the units of Planck’s constant were the same as those of angular momentum and so he reasoned that perhaps Planck’s constant was a measure of the angular momentum of the orbiting electron. But Nicholson went one step further and argued that the angular momentum of the orbiting electron could take on values which were an integer multiple of Planck’s constant. This allowed Bohr to develop a model in which the differences between the energy levels matched those of the empirically developed Rydberg formula. When the Bohr model was superseded Nicholson’s assumption was simply carried forward unchallenged into these later models.
The main problem with Nicholson’s assumption is that it lacks any mathematical rigour. It simply takes one variable, angular momentum, and asserts that if we allow it to have this characteristic quantisation then we get energy levels which appear to be correct. In so doing it fails to provide any sort of explanation as to why such a quantisation should take place.
In the 1940s a branch of mathematics appeared which straddles the boundary between continuous functions and discrete solutions. It was developed by engineers at Bell Labs to address problems with capacity in the telephone network. While at first site there appears to be little to connect problems of network capacity with electrons orbiting atomic nuclei it is the application of these mathematical ideas which holds the key to explaining quantisation inside the atom.

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